Geodesic
Generating elements for groups and Lie algebras of the form $F/[N,N]$
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 2, pp. 60-71

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Let $F$ be a free product of groups $A_i~(i\in I)$ and a free group $G$ and its normal subgroup $N$ has trivial intersection with each factor $A_i$. Subject to these conditions we will establish necessary and sufficient conditions for an element of the group $F/[N,N]$ belongs to the subgroup generated by a given finite set of elements of $F/[N,N]$ and necessary and sufficient conditions for a given set of elements of the group $F/[N,N]$ to generate it. Similar results are proved also for Lie algebras.
Keywords: group ring, Lie algebra, universal enveloping algebra. 
@article{VNGU_2015_15_2_a4,
     author = {A. F. Krasnikov},
     title = {Generating elements for groups and {Lie} algebras of the form $F/[N,N]$},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {60--71},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2015_15_2_a4/}
}
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A. F. Krasnikov. Generating elements for groups and Lie algebras of the form $F/[N,N]$. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 2, pp. 60-71. http://geodesic.mathdoc.fr/item/VNGU_2015_15_2_a4/